Does rotations acts on spin?

Question

1. Rotations are part of the symmetries of nature
2. I heard that electrons carry a spin 1/2

Q: How do rotations acts on the spin of an electron?

The state of my reflection: irreducible representations of rotations are dimension odd. But the spin is dimension 2, so it is either invariant under rotation, either incomplete (there must be another physical quantity that put together with the spin can be acted by rotations). update: after discussing with a friend, I think now that the spin state is not a representation of SO(3) but the observables (the spin operators) are. My answer so far is: No

I hope my question is clear, it's a physics question, this is not a question about group theory, I know what is SO(3) and SU(2) and their relationship.

Answer 1

The transformation properties of a quantum state under rotations are not described by representations of the rotation group $SO(3)$ but rather by represetations of its double cover $SU(2)$. So, a spin $\frac{1}{2}$ particle is absolutely acted upon by rotations but not under a representation of $SO(3)$ but rather under a two dimensional irreducible representation of $SU(2)$.

Answer 2

To have a more direct answer, this is a question whose answer is actually calculable. Since you are arguing with group theory I assume that you know something about relativistic quantum field theory.

Spin 1/2 particles are the one particle states generated by the Dirac Spinor field, which is described by the lagrangian $\bar{\psi}(i\partial_{\mu}\gamma^{\mu}-m)\psi$.

A Lorentz transformation $\lambda$ in this representation can be written as $U(\lambda) = exp(-i\omega_{\mu \nu}S^{\mu \nu}), S^{\mu \nu} = i/4 [\gamma^{\mu},\gamma^{\nu}]$ Which acts on $\psi$ as $\psi' = U(\lambda)\psi$.

Rotations are generated by the elements $\omega_{ij}, i,j= 1,2,3$. One finds the infinitesimal transformation around e.g. the z axis around an angle $\theta = \omega_{12}=-\omega_{21}$ to be $U = 1-\theta i/2 \Sigma^3, \Sigma^3 = diag(1,-1,1,-1)$ in Weyl representation of the $\gamma$ matrices. So the action on a spinor will change its components, which shows that rotations act on spin also.